Polynomial Notes

RVC MTH 120 College Algebra Chapter 5 Notes Synthetic Division – an easy way to divide higher-degree polynomials by binomials of the form x – r Zero of a polynomial P(x) – any number r for which P(r) = 0 Zeros of the polynomial P(x) are real roots of the polynomial equation P(x) = 0 Remainder Theorem – If P(x) is divided by x – r, the remainder is P(r) Factor Theorem – If r is a zero of P(x), then x – r is a factor of P(x) or If x – r is a factor of P(x) then r is a zero of P(x) Fundamental Theorem of Algebra – If P(x) is a polynomial with positive degree, then P(x) has at least one zero Root of Multiplicity k – any number r that occurs k times as a root of a polynomial equation Conjugate Pairs Theorem – If a polynomial equation with real-number coefficients has a complex root a + bi with b ≠ 0, then its conjugate a – bi is also a root Descartes’ Rule of Signs – Given a polynomial with real coefficients, the number of positive roots of the polynomial is either equal to the number of variations in sign of the polynomial or less than that by an even number The number of negative roots is either equal to the number of variations in the sign of P(-x) or less than that by an even number Upper Bound – Let the lead coefficient of a polynomial with real coefficients be positive, and do a synthetic division of the coefficients by a positive number c. If each term in the last row of the division is nonnegative, no number greater than c can be root of the polynomial Lower Bound - Let the lead coefficient of a polynomial with real coefficients be positive, and do a synthetic division of the coefficients by a negative number d. If each term in the last row of the division alternate signs, no number less than d can be root of the polynomial. If 0 appears in the last row, that 0 can be assigned either a + or a – sign to help signs alternate